The DFT

Preface List of Frequently Used Terms Chapter 1: Introduction. A Bit of History An Application Problems Chapter 2: The Discrete Fourier Transform . Introduction DFT Approximation to the Fourier Transform The DFT-IDFT Pair DFT Approximations to Fourier Series Coefficients The DFT from Trigonometric Approximation Transforming a Spike Train Limiting Forms of the DFT-IDFT Pair Problems Chapter 3: Properties of the DFT. Alternate Forms for the DFT Basic Properties of the DFT Other Properties of the DFT A Few Practical Considerations Analytical DFTs Problems Chapter 4: Symmetric DFTs. Introduction Real Sequences and the Real DFT (RDFT) Even Sequences and the Discrete Cosine Transform (DST) Odd Sequences and the Discrete Sine Transform (DST) Computing Symmetric DFTs Notes Problems Chapter 5: Multidimensional DFTs. Introduction Two-Dimensional DFTs Geometry of Two-Dimensional Modes Computing Multi-Dimensional DFTs Symmetric DFTs in Two Dimensions Problems Chapter 6: Errors in the DFT Introduction Periodic, Band-Limited Input Periodic, Non-Band-Limited Input Replication and the Poisson Summation Formula Input with Compact Support General Band-Limited Functions General Input Errors in the Inverse DFT DFT Interpolation - Mean Square Error Notes and References Problems Chapter 7: A Few Applications of the DFT. Difference Equations - Boundary Value Problems Digital Filtering of Signals FK Migration of Seismic Data Image Reconstruction from Projections Problems Chapter 8: Related Transforms. Introduction The Laplace Transform The z-Transform The Chebyshev Transform Orthogonal Polynomial Transforms The Discrete Hartley Transform (DHT) Problems Chapter 9: Quadrature and the DFT. Introduction The DFT and the Trapezoid Rule Higher Order Quadrature Rules Problems Chapter 10: The Fast Fourier Transform. Introduction Splitting Methods Index Expansions (One ---> Multi-dimensional) Matrix Factorizations Prime Factor and Convolution Methods FFT Performance Notes Problems Appendix: Table of Discrete Fourier Transforms Bibliography.